Calculus Examples

$6 \log_{3} (x - 2)$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $6 \log_{3} (x - 2)$ with respect to $x$ is $6 \frac{d}{d x} [\log_{3} (x - 2)]$ .

$6 \frac{d}{d x} [\log_{3} (x - 2)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{3} (x)$ and $g (x) = x - 2$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x - 2$ .

$6 (\frac{d}{d u} [\log_{3} (u)] \frac{d}{d x} [x - 2])$

Step 2.2

The derivative of $\log_{3} (u)$ with respect to $u$ is $\frac{1}{u \ln (3)}$ .

$6 (\frac{1}{u \ln (3)} \frac{d}{d x} [x - 2])$

Step 2.3

Replace all occurrences of $u$ with $x - 2$ .

$6 (\frac{1}{(x - 2) \ln (3)} \frac{d}{d x} [x - 2])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{(x - 2) \ln (3)}$ and $6$ .

$\frac{6}{(x - 2) \ln (3)} \frac{d}{d x} [x - 2]$

Step 3.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{6}{(x - 2) \ln (3)} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{6}{(x - 2) \ln (3)} (1 + \frac{d}{d x} [- 2])$

Step 3.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{6}{(x - 2) \ln (3)} (1 + 0)$

Step 3.5

Simplify by multiplying through.

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Step 3.5.1

Add $1$ and $0$ .

$\frac{6}{(x - 2) \ln (3)} \cdot 1$

Step 3.5.2

Multiply $\frac{6}{(x - 2) \ln (3)}$ by $1$ .

$\frac{6}{(x - 2) \ln (3)}$

Step 3.5.3

Apply the distributive property.

$\frac{6}{x \ln (3) - 2 \ln (3)}$